Our driving force will be a class of probability distributions on the \(d\) -dimensional box \(B_n^d = \lbrace 0,1/n,2/n,\ldots ,1\rbrace ^d\)  that exhibit this problem and show that a very fast and very simple simulated annealing scheme yet provides convergence to the true distribution within the order of \(n^2\)  steps. Furthermore, combining simulated annealing with a varying step size, this can even be taken down to order \(n \log n\) . This class comprises an abundance of interesting examples, of which we will include the mean field Ising model with a nonzero external field, collapsed Gibbs sampling for Latent Dirichlet allocation (LDA) and a model of nuclear physics; calibration data corresponding to the 3S1 phase shifts from an analysis of neutron-proton scattering cross sections [1]}. In all of these cases, we have run a series of experiments, of which the LDA example are large scale whereas the others are on a smaller scale.
